Lt. Gov. Dan Patrick (right) seen with Gov. Greg Abbott, on Jan. 20, wrote “Wake up, Mr. President!” on Facebook regarding a report that two Syrian families had tried to enter the country at the Texas-Mexico border.

Lt. Gov. Dan Patrick (right) seen with Gov. Greg Abbott, on Jan....

AUSTIN — Fear that terrorists could gain entry to the United States by posing as refugees reached frenzied levels Thursday, as a report that two Syrian families walked up to a Texas-Mexico border crossing increased already-heated rhetoric from worried state officials and presidential hopefuls.

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“Wake up, Mr. President!” demanded Lt. Gov. Dan Patrick in a Facebook post. “ISIS maybe?” wondered Donald Trump on Twitter, adding, “I told you so.” Both messages said the Syrians were “caught,” the terminology used in the original report by the conservative Breitbart News, which had been shared more than 116,000 times on Facebook as of Thursday evening.

In reality, according to the Homeland Security Department, the two men, two women and four children “presented themselves at a port of entry in Laredo,” presumably to ask for asylum, which happens every day with people from all over the world. Like many asylum seekers, the Syrians will be taken to detention centers as officials assess their backgrounds and needs.

It is rare for border officials to apprehend Syrians illegally crossing the border who do not turn themselves in: It happened just twice in 2013, 14 times in 2014 and five times since Oct. 1, according to government data.

Immigration experts pushed back Thursday on the reaction to the possibility of terrorists posing as Syrian refugees.

“Obviously, people are very sensitive to the security threat that unknown people arriving at the border from Syria and other countries in that part of the world represent right now,” said Marc Rosenblum, deputy director of immigration policy at the national nonprofit Migration Policy Institute. “But the circumstances under which these people presented themselves, they don’t meet the profile of a threat. These families arrived and surrendered to the first border officer they could find.”

Alex Nowrasteh, an immigration policy analyst with the national Cato Institute, said the incident “shows the asylum system and border security are working well because these folks decided to turn themselves in.”

Other analysts said the dust-up served as a symbol of the growing polarization between U.S. residents in the wake of last week’s deadly terrorist attacks in Paris, as liberals and conservatives increasingly appear to be operating in different realities about the threat of those fleeing the war-torn Middle East, the vulnerability of America’s refugee, asylum and border security systems, and the legal rules governing it all.

On Thursday, some of the division seemed to stem from confusion about the differences between refugees and asylum seekers.

Refugees are processed overseas by the United Nations and then granted admission to the United States — after waiting an average of two years and undergoing a series of background checks. Asylum seekers, on the other hand, generally turn themselves in to authorities after arriving here and expressing fear of returning home. They, too, are subject to a series of background checks.

To be granted permission to stay, both must show that they have a well-founded fear of persecution if they return home because of their race, religion, nationality, membership in a social group or political affiliation.

In the fiscal year that ended Sept. 30, nearly 1,700 Syrian refugees were resettled in the nation overall, according to the State Department.

This fall, President Barack Obama raised the ceiling for total annual refugee admissions from 70,000 to 85,000, said it would increase to 100,000 by 2017, and pledged to accept 10,000 Syrians in the next fiscal year.

The latter policy has drawn criticism in the days since terrorists thought to have ties to the Syria-based Islamic State killed at least 129 people with guns and suicide bombs. More than 30 governors, including Texas’ Greg Abbott, have said their states will not accept refugees fleeing the country.

Abbott laid out his legal justification for refusing refugees for the first time late Wednesday, telling FOX News host Sean Hannity that Section 1522 of Title 8 of the U.S. Code says all activities by local refugee resettlement agencies “should be conducted in close cooperation and advance consultation with state and local governments.”

Several legal experts disagreed with the governor Thursday, saying the phrase does not change the reality that on immigration matters, federal policy trumps state preference. The federal government, they said, can place refugees from any country with any local resettlement agency, regardless of what the state says.

“The sole authority to place refugees is federal,” said Michael Olivas, a University of Houston law professor who specializes in immigration.